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By applying computer algebra tools (mainly, Maple and C++), given the Dynkin diagram ∆ = A n , with n ≥ 2 vertices and the Euler quadratic form q ∆ : Z n → Z, we study the problem of classifying mesh root systems and the mesh geometries of roots of ∆ (see Section 1 for details). The problem reduces to the computation of the Weyl orbits in the set Mor ∆ ⊆ M n (Z) of all matrix morsifications A of q ∆ , i.e., the non-singular matrices and (ii) the Coxeter matrix Cox the Coxeter number), and the Coxeter polynomial coxThe problem of determining the W ∆ -orbits Orb(A) of Mor ∆ and the Coxeter polynomials cox A (t), with A ∈ Mor ∆ , is studied in the paper and we get its solution for n ≤ 8, and A = [a ij ] ∈ Mor An , with |a ij | ≤ 1. In this case, we prove that the number of the W ∆orbits Orb(A) and the number of the Coxeter polynomials cox A (t) equals two or three, and the following three conditions are equivalent: (i)We also construct: (a) three pairwise different W ∆ -orbits in Mor ∆ , with pairwise different Coxeter polynomials, if ∆ = A 2m−1 and m ≥ 3; and (b) two pairwise different W ∆ -orbits in Mor ∆ , with pairwise different Coxeter polynomials, if ∆ = A 2m and m ≥ 1.

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